Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group


Authors: D. Danielli, N. Garofalo and D. M. Nhieu
Journal: Proc. Amer. Math. Soc. 140 (2012), 811-821
MSC (2010): Primary 49Q05; Secondary 53D10
DOI: https://doi.org/10.1090/S0002-9939-2011-11058-X
Published electronically: November 2, 2011
MathSciNet review: 2869066
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of the local summability of the sub-Riemannian mean curvature $ \mathcal H$ of a hypersurface $ M$ in the Heisenberg group, or in more general Carnot groups, near the characteristic set of $ M$ arises naturally in several questions in geometric measure theory. We construct an example which shows that the sub-Riemannian mean curvature $ \mathcal H$ of a $ C^2$ surface $ M$ in the Heisenberg group $ \mathbb{H}^1$ in general fails to be integrable with respect to the Riemannian volume on $ M$.


References [Enhancements On Off] (What's this?)

  • [B] Z. M. Balogh, Size of characteristic sets and functions with prescribed gradients,
    J. Reine Angew. Math., 564 (2003), 63-83. MR 2021034 (2005d:43007)
  • [CHMY] J. H. Cheng, J. F. Hwang, A. Malchiodi and P. Yang, Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group, revised version, 2004, Ann. Sc. Norm. Sup. Pisa, 4 (2005), 129-177. MR 2165405 (2006f:53008)
  • [DGN1] D. Danielli, N. Garofalo and D. M. Nhieu, Sub-Riemannian calculus on hypersurfaces in Carnot groups, Advances in Math., 215 (2007), no. 1, 292-378. MR 2354992 (2009h:53061)
  • [DGN2] -, A partial solution of the isoperimetric problem for the Heisenberg group, Forum Math., 20 (2008), no. 1, 99-143. MR 2386783 (2009j:53030)
  • [DGN3] -, A sub-Riemannian Minkowski formula, in preparation.
  • [HP] R. K. Hladky and S. D. Pauls, Variation of perimeter measure in sub-Riemannian geometry, preprint, 2007.
  • [Ma] V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc. (JEMS), 8 (2006), no. 4, 585-609. MR 2262196 (2007i:53032)
  • [Mo] F. Montefalcone, Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups, preprint, 2009.
  • [RR] M. Ritoré and C. Rosales, Area-stationary surfaces in the Heisenberg group $ \mathbb{H}^1$, Adv. Math., 219 (2008), no. 2, 633-671. MR 2435652 (2009h:49075)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 49Q05, 53D10

Retrieve articles in all journals with MSC (2010): 49Q05, 53D10


Additional Information

D. Danielli
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: danielli@math.purdue.edu

N. Garofalo
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: garofalo@math.purdue.edu

D. M. Nhieu
Affiliation: Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan, Republic of China
Email: dmnhieu@math.ncu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-2011-11058-X
Keywords: Minimal surfaces, $H$-mean curvature, integration by parts, first and second variation, monotonicity of the $H$-perimeter
Received by editor(s): August 25, 2010
Received by editor(s) in revised form: August 31, 2010
Published electronically: November 2, 2011
Additional Notes: The first author was supported in part by NSF grant CAREER DMS-0239771
The second author was supported in part by NSF Grant DMS-1001317
The third author was supported in part by NSC Grant 99-2115-M-008-013-MY3
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society